What is the value of F(-1)-F(1)?

Question

GMATH  practice exercise (Quant Class 14)

(1) F(x) = x^2 , for all x 
(2) F(x+1) = F(x) + 2x + 1, for all x

chetan2u · Answer

So we are looking for  F(-1)-F(1) .

(1)  F(x) = x^2  , for all x 
We can get the value of F(-1) and F(1), so sufficient
 F(-1) = (-1)^2=1  
 F(1) = (1)^2=1  
so  F(-1)-F(1)=1-1=0 
Sufficient

(2)  F(x+1) = F(x) + 2x + 1 , for all x
The difference of 1 in x and x+1 should tempt you to rearrange and work on the variables accordingly..
Now F(-1) and F(1) have difference of 1-(-1) or 2, so let us work in similar fashion..
  F(x+1) = F(x) + 2x + 1......... F(x)-F(x+1) = -( 2x + 1) 
a) let x = -1, so  F(x)-F(x+1) = -( 2x + 1)......F(-1)-F(-1+1) = -( 2(-1) + 1)=-(-1)=1 , that is  F(-1)-F(0) = 1 
b) let x = 0, so  F(x)-F(x+1) = -( 2x + 1)......F(0)-F(0+1) = -( 2(0) + 1)=-(1)=-1 , that is  F(0)-F(1) = -1 
Add the two colored equation F(-1)-F(0)+ F(0)-F(1) = 1+ (-1)..........
so  F(-1)-F(1)=1-1=0 
Sufficient

D

prabsahi · Answer

This approach is really good and I got struck at the highlighted point.
 
Here is my approach. The main concern is statement 2

1.F(x) = x^2 , for all x

F(-1) = x^2 = 1
F(1)= x^2 = 1

So we have F(-1) - F(1)= 0

2.
F(x+1) = F(x) + 2x + 1, for all x ---------------------- Main equation

Now my line of thinking was:Lets get F(-1) so I did x+1=-1 which gave x=-2.
Now substituting in main equation 2

F(-2+1)= F(-2) + 2*(-2) +1

=> F(-1) = F(-2) - 4 + 1
 =F(-2) -3

I think this is wrong step,I identified a little later.

We should rather take X=-1 directlly ,this will save us from getting F(-2) and have equation in F(0) form thereby saving us from other variable.

so substituing X=-1 in main equation

F(0) = F(-1) + 2*-1 +1
 = F(-1) - 1 ---------------------------------- 1

Again to get F(1) Lets make x+1= 1 ,X=0
Now again substituting in main equation 2

F(1) = F(0) + 2*0 + 1
 =F(0) + 1 --------------------2
implies

F(0) = F(1)-1

Substituting F(0) in equation 1

F(1)-1 = F(-1) - 1

Rearranging

F(-1) - F(1) = 0

Hence sufficient.

Hope it helps!!

fskilnik · Answer

? = F\left( { - 1} \right) - F\left( 1 \right)

\left( 1 \right)\,\,F\left( x \right) = {x^2}\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\, \Rightarrow \,\,\,\,F\left( x \right) = F\left( { - x} \right)\,,\,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\,\mathop \Rightarrow \limits^{x\, = \,1} \,\,\,\,\,F\left( 1 \right) = F\left( { - 1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 0

\left( 2 \right)\,\,F\left( {x + 1} \right) - F\left( x \right) = 2x + 1\,,\,{\rm{for}}\,\,{\rm{all}}\,\,x\,\,\,\left( * \right)

\left. \matrix{
 {\rm{Take}}\,\,x = - 1\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 0 \right) - F\left( { - 1} \right) = - 2 + 1\,\,\, \hfill \cr 
 {\rm{Take}}\,\,x = 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,F\left( 1 \right) - F\left( 0 \right) = 1 \hfill \cr} \right\}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,F\left( 1 \right) - F\left( { - 1} \right) = 1 + \left( { - 2 + 1} \right) = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,? = - 0 = 0

The correct answer is therefore (D).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

What is the value of F(-1)-F(1)?

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